In ordinary steady flight, the airplane must develop enough upward
force to support its weight, i.e. to counteract the downward force of
gravity. It is a defining property of an aircraft (as opposed to a
ballistic missile, spacecraft, or watercraft) that virtually all of
this upward force comes from the air.

Whenever the
airplane applies a downward force to the air, the air applies an equal
amount of upward force to the airplane.

Idea 1 is the cornerstone of any understanding of how
the airplane is able to fly. It is 100% true. It is a direct
consequence of a fundamental principle of physics, namely conservation
of momentum; for details on this, see section 19.2.
Everything else in this chapter is just a clarification or an
elaboration of this simple idea. For example, we shall see in
section 3.3 that it is better to think of the wing as
pulling down on the air, rather than pushing.

As always, whenever you come across a new idea, you should mull it
over, checking to see how it connects – or conflicts – with
other things you know. See section 21.10 for more on this. In
this case:

Everybody knows
that if you try to push on the air with your hand, the air moves out
of the way before you can develop much force.

To reconcile idea 1 with idea 2, observe that
the airplane is moving sideways, so that at each moment, it is pulling
down on a new parcel of air. It transfers some momentum before
the parcel has time to move out of the way. Combining these two ideas
allows us to make a prediction:

In this chapter I will explain a few things about how air behaves as
it flows past a wing. Many of the illustrations – such as figure 3.1 – were produced by a wind-tunnel simulation1 program that I wrote for my computer. The wing is
stationary in the middle of the wind tunnel; air flows past it from
left to right. A little ways upstream of the wing (near the left edge
of the figure) I have arranged a number of smoke injectors. Seven of them are on all the time, injecting
thin streams of purple smoke. The smoke is carried past the wing by
the airflow, making visible stream
lines.

In addition, on a five-times closer vertical spacing, I inject
pulsed streamers. The smoke is turned on for 10
milliseconds out of every 20. In the figure, the blue smoke was
injected starting 70 milliseconds ago, the green smoke was injected
starting 50 milliseconds ago, the orange smoke was injected starting
30 milliseconds ago, and the red smoke was injected starting 10
milliseconds ago. The injection of the red smoke was ending just as
the snapshot was taken.

Figure 3.2 points out some important properties of
the airflow pattern. For one thing, we notice that the air just ahead
of the wing is moving not just left to right but also upward; this is
called upwash. Similarly, the air just aft of the wing is
moving not just left to right but also downward; this is called
downwash. Downwash behind the wing is relatively easy to
understand; the whole purpose of the wing is to impart some downward
motion to the air.

The upwash in front of the wing is a bit more interesting. As
discussed in section 3.8, air is a fluid, which means it can
exert pressure on itself as well as other things. The air pressure
strongly affects the air, even the air well in front of the wing.
Upwash and downwash are discussed in more detail in section 3.4.

Along the leading edge of the wing there is something called a
stagnation line, which is the dividing line between air that
flows over the top of the wing and air that flows under the bottom of
the wing. On an airplane, the stagnation line runs the length of the
wingspan, but since figure 3.2 shows only a cross
section of the wing, all we see of the stagnation line is a single
point.

Another stagnation line runs spanwise along the trailing
edge. It marks the place where air that passed above the wing
rejoins air that passed below the wing.

We see that at moderate or high angles of attack, the forward
stagnation line is found well below and aft of the leading
edge of the wing. The air that meets the wing just above the
stagnation line will backtrack toward the nose of the airplane, flow
up over the leading edge, and then flow aft along the top of the wing.

The set of all points that passed the injector array at a given time
defines a timeline. The right-hand edge of the orange smoke is
the “30 millisecond” timeline. Several of the timelines are labeled
according to their age in figure 3.3.

Since the air near the wing is flowing at all sorts of different
speeds and directions, the question arises of what is the “true”
airspeed in the wind tunnel. The logical thing to do is to measure
the velocity of the free stream; that is,
at a point well upstream, before it has been disturbed by the wing.

The pulsed streamers give us a lot of information. Regions where the
pulsed streamers have been stretched out are high velocity regions.
This is pretty easy to see; each pulsed streamer lasts exactly 10
milliseconds, so if it covers a long distance in that time it must be
moving quickly. The maximum velocity produced by this wing at this
angle of attack is approximately twice the free-stream velocity.
Airfoils can be very effective at speeding up the air.

Conversely, regions where the pulsed streamers cover a small distance
in those 10 milliseconds must be low-velocity regions. The minimum
velocity is zero. That occurs near the front and rear stagnation
lines.

The relative wind vanishes on the stagnation lines.
A small bug walking on the wing of an airplane in flight could
walk along the stagnation line without feeling any wind.2

Stream lines have a remarkable property: the air
can never cross a stream line. That is because of the way the stream
lines were defined: by the smoke. If any air tried to flow past a
point where the smoke was, it would carry the smoke with it.
Therefore a particular parcel of air bounded by a pair of stream lines
(above and below) and a pair of timelines (front and rear) never loses
its identity. It can change shape, but it cannot mix with another
such parcel.3

Another thing we should
notice is that in low velocity regions, the stream lines are
farther apart from each other. This is no accident. At
reasonable airspeeds, the wing doesn’t push or pull on the air hard
enough to change its density significantly (see
section 3.6.3 for more on this). Therefore the air
parcels mentioned in the previous paragraph do not change in area when
they change their shape. In one region, we have a long, skinny parcel
of air flowing past a particular point at a high velocity. (If the
same amount of fluid flows through a smaller region, it must be
flowing faster.) In another region, we have a short fat parcel
flowing by at a low velocity.

The most remarkable thing about this figure is that the blue smoke
that passed slightly above the wing got to the trailing edge 10 or 15
milliseconds earlier than the corresponding smoke that passed
slightly below the wing.

This is not a mistake. Indeed, we shall see in section 3.12.3
that if this were not true, it would be impossible for the wing to
produce lift.

This may come as a shock to many readers, because all sorts of
standard references claim that the air is somehow required to pass
above and below the wing in the same amount of time. I have seen this
erroneous statement in elementary-school textbooks, advanced physics
textbooks, encyclopedias, and well-regarded pilot training handbooks.
Bear with me for a moment, and I’ll convince you that figure 3.3 tells the true story.

First, I must convince you that there is no law of
physics that prevents one bit of fluid from being delayed relative
to another.

Consider the scenario depicted in figure 3.4. A river of water
is flowing left to right. Using a piece of garden hose, I siphon some
water out of the river, let it waste some time going through several
feet of coiled-up hose, and then return it to the river. The water
that went through the hose will be delayed. The delayed parcel of
water will never catch up with its former neighbors; it will not even
try to catch up.

Note that delaying the water did not require
compressing the water, nor did it require
friction.

Let’s now discuss the behavior of air near a wing. We will see that
there are two parts to the story: The obstacle effect, and the
circulation effect.

The first part of the story is that the wing is an obstacle to the
air. Air that passes near such an obstacle will be delayed. In fact,
air that comes arbitrarily close to a stagnation line will be
delayed an arbitrarily
long time. The air molecules just hang around in the vicinity of the
stagnation line, like the proverbial donkey midway between two bales
of hay, unable to decide which alternative to choose.

Air near the wing is delayed relative to an undisturbed parcel of air.
The obstacle effect is about the same for a parcel passing above the
wing as it is for the parcel passing a corresponding distance below
the wing. This effect falls off very quickly as a function of
distance from the wing. You can see that the air that hits the
stagnation line dead-on (the middle blue streamer) never makes it to
the trailing edge, as you can see in all three panels of figure 3.5.
When the wing is producing zero lift, this obstacle effect is pretty
much the whole story, as shown in the top panel of figure 3.5.

Now we turn to the second part of the story, the circulation effect.
In figure 3.5 the three panels are labelled as to angle of attack.
Lift is proportional to angle of attack whenever the angle is not too
large. In particular, the zero-lift case is what we are calling zero
angle of attack, even for cambered wings, as discussed in
section 2.2.

For the rest of this section, we assume the wing is producing a
positive amount of lift. This makes the airflow patterns much more
interesting, as you can see from the second and third panels of
figure 3.5. An air parcel that passes above the wing arrives at the
trailing edge early. It arrives early compared to the parcel a
corresponding distance below the wing, with no exceptions. This is
because of something called circulation, as will be discussed in
section 3.12.

We can also see that most of the air passing above the wing
arrives early in absolute terms, early compared to an undisturbed
parcel of air. The exception occurs very close to the wing, where the
obstacle effect (as previously discussed) overwhelms the circulation
effect.

Unlike the obstacle effect, the circulation effect drops off quite
slowly. It extends for quite a distance above and below the wing – a
distance comparable to the wingspan.

A wing is amazingly effective at producing circulation, which speeds
up the air above it. Even though
the air that passes above the wing has a longer path, it gets to the
back earlier than the corresponding air that passes below the
wing.

Note the contrast:

The change in speed is temporary. As the
air reaches the trailing edge and thereafter, it quickly returns to
its original, free-stream velocity (plus a slight downward component).
This can been seen in the figures, such as figure 3.3
— the spacing between successive smoke pulses returns to its
original value.

The change in relative position is permanent. If we
follow the air far downstream of the wing, we find that the air that
passed below the wing will never catch up with the corresponding air
that passed above the wing. It will not even try to catch up.

Figure 3.6 is a contour plot that shows what the pressure
is doing in the vicinity of the wing. All pressures will be measured
relative to the ambient atmospheric pressure in the free stream. The
blue-shaded regions indicate suction, i.e. negative pressure relative
to ambient, while the red-shaded regions indicate positive pressure
relative to ambient. The dividing line between pressure and suction
is also indicated in the figure.

Note on units: The pressure and suction near the
wing are conveniently measured in multiples of the dynamic
pressure,4Q. In figures such as
figure 3.6, each contour represents exactly 0.2 Q. We
choose units of Q, rather than more prosaic units such as
PSI, because it allows the figure to remain quantitatively accurate
over a rather wide range of airspeed and density conditions. If you
know the dynamic pressure, you can figure out what the wing
is doing; you don’t need to know the airspeed or
density separately.

As a numerical example: If you are doing 100 knots under standard sea
level conditions, we have:

Whenever we are talking about pressure in connection with lift and drag,
it is safe to assume we mean gauge pressure, i.e. pressure
relative to the ambient free-stream pressure – not absolute
pressure – unless the context clearly demands otherwise. Ordinary
light-aircraft speeds are small compared to the speed of sound,
which guarantees that the dynamic pressure Q is always small
compared to 1 Atm. Therefore if you hear somebody talking
about a pressure on the order of 1Q, you know it
must be gauge pressure, not absolute pressure. Furthermore
it should go without saying that any mention of suction
refers to gauge pressure, since there is no such thing
as negative absolute pressure.

The maximum positive pressure on any airfoil is exactly equal to Q.
This occurs right at the stagnation lines. This stands to reason,
since by Bernoulli’s principle, the slowest air has the highest
pressure. At the stagnation lines, the air is stopped — which is
slow as it can get. See section 3.6, especially
figure 3.10.

The maximum suction near an airfoil depends on the angle of attack,
and on the detailed shape of the airfoil. Similarly-shaped airfoils
tend to exhibit broadly similar behavior. By way of example, the
angle of attack in figure 3.6 is 3 degrees, a reasonable
“cruise” value. For this airfoil under these conditions, the max
suction is just over 0.8 Q.

There is a lot we can learn from studying this figure. For one thing,
we see that the front quarter or so of the wing does half of the
lifting, which is typical of general-aviation airfoils. That means
the wing produces relatively little pitch-wise torque around the
so-called “quarter chord” point. This is why engineers typically
put the main wing spar at or near the quarter chord point. Another
thing to notice is that suction acting on the top of the wing is
vastly more important than pressure acting on the bottom of the wing.

For the airfoil in figure 3.6, under cruise conditions, there
is almost no high pressure on the bottom of the wing; indeed there is
mostly suction there.5 The only reason the wing can support
the weight of the airplane is that there is more suction on the
top of the wing. (There is a tiny amount of positive pressure on the
rear portion of the bottom surface, but the fact remains that suction
above the wing does more than 100% of the job of lifting the
airplane.)6

This pressure pattern would be really hard to explain in terms of
bullets bouncing off the wing. Remember, the air is a fluid, as
discussed in section 3.8. It has a well-defined pressure
everywhere in space. When this pressure field meets the wing, it
exerts a force: pressure times area equals force.

At higher angles of attack, above-atmospheric pressure does develop
below the wing, but it is always less pronounced than the
below-atmospheric pressure above the wing.

Both upwash and downwash make a positive contribution to lift. This
can be understood as follows. In figure 3.7, the plank
plays the role of the airplane. We imagine you are riding on the
plank. Now suppose somebody on the ground throws a baseball upwards.
This is upwash. When the baseball hits the plank, it exerts an upward
force on the plank. If you caught the baseball at this point, that
would be the end of the story. You would be getting lift from upwash
alone.

Next, suppose that after catching the ball, you throw it downwards.
This is downwash. This provides additional upward force, i.e.
additional lift.

In fact, you don’t need to catch or throw the ball; all you need to do
is let it bounce off the bottom of the plank. Each bounce provides
two units of upward impulse, one from the incoming upwash and one from
the outgoing downwash.

Both the upwash and the downwash are related to the circulation in the
air, as discussed in section 3.12. Lift requires
circulation and vorticity, as discussed in section 3.14.

Beware: It is a common mistake to think that upwash exists because the
wing is sucking upward on the air. Such suction would cause a
downward force on the wing, making a negative contribution to lift.
In fact the upwash flows uphill, into a region of high and increasing
pressure, slowing down as it goes. You can see this in
figure 3.8, in the region framed by the purple rectangle, just
below and ahead of the wing. Bottom line: Both the upwash and the
downwash make a positive contribution to the lift.

Let’s consider the relationship between pressure and velocity. It
turns out that given the velocity field, it is rather straightforward
to calculate the pressure field. Indeed there are two ways to do
this; we discuss one of them here, and the other in section 3.6.

Figure 3.9 shows the pressure and velocity at various angles of
attack. The velocity distribution depends on angle of attack, and so
does the pressure distribution.

We know that air has mass. Moving air has momentum. If the air
parcel follows a curved path, there must be a net force on it, as
required by Newton’s laws.7

Pressure alone does not make a net force; you need a pressure
difference so that one side of the air parcel is being pressed
harder than the other. Therefore the rule is this: If at any place
the stream lines are curved, the pressure at nearby places is
different.

You can see in the figures that tightly-curved streamlines correspond
to big pressure gradients and vice versa.

If you want to know the pressure everywhere, you can start somewhere
and just add up all the changes as you move from place to place to
place. This is mathematically tedious, but it works. It works even
in situations where Bernoulli’s principle isn’t immediately applicable.

We now discuss a second way in which pressure is related to velocity,
namely Bernoulli’s principle, aka Bernoulli’s formula. In situations
where this formula can be applied (which includes most situations –
but not all), this is by far the slickest way of doing things.

Bernoulli’s principle is very easy to understand provided the
principle is correctly stated. However, we must be careful, because
seemingly-small changes in the wording can lead to completely wrong
conclusions.

For simplicity, let’s consider a scenario where you are sitting in the
airplane, in flight. We restrict attention to situations where the
effects of friction can be neglected. We will analyze the same
situation in two different ways.

First analysis: We pick a particular location in your reference frame,
located at some fixed distance relative to you. As a premise of the
scenario, we assume the air pressure, velocity, density, etc. at this
location are constant. If you measure things at this location now,
and come back and measure them again later, everything is the same.
We call this a steady flow situation.

Second analysis: Rather than considering a particular location in
space, we ask what happens to a particular parcel of air as it flows
along a streamline. Even though the properties of pressure, velocity,
density, etc. that pertain to a particular location are not changing,
the properties that pertain to a particular parcel of air will change
as the parcel flows from location to location.

We will now state the general idea of Bernoulli’s principle.
In this scenario, for any particular parcel of fluid:

The explanation for this principle is completely logical and
straightforward: The idea is that as the parcel moves along, following
a streamline, as it moves into an area of higher pressure there will
be higher pressure ahead (higher than the pressure behind) and this
will exert a force on the parcel, slowing it down. Conversely if the
parcel is moving into a region of lower pressure, there will be an
higher pressure behind it (higher than the pressure ahead), speeding
it up. As always, any unbalanced force will cause a change in
momentum (and velocity), as required by Newton’s laws of motion.

There are various ways of quantifying this idea, depending on what
sort of simplifications and approximations you want to make. Suppose
we have two points B and A (denoting “before” and “after”) not
too far apart. We continue to neglect viscosity and to assume steady
flow. Then we can describe the flow of a single parcel of air as
follows:

where P denotes pressure, v denotes airspeed, and ρ denotes
the density, i.e. mass per unit volume. In general ρA will be
different from ρB but we are not going to worry about it for the
moment, because the whole equation is only valid to first order, and
worrying about ρA − ρB would be a second-order correction.

We must be careful, because the constant in
equation 3.5 is only constant to first order.
Indeed, the whole equation is only valid to first order. That is,
the equation is only valid when the pressure P is close to the
ambient pressure and the actual density ρ is close to the
ambient density. (If you want to understand how close, see
equation 3.7 and/or or equation 3.8.)

This means that if we are going to invoke the simplifications
and approximations that lead to
equation 3.5, the velocity vA must be close to the
velocity vB, or that both must be small compared to the speed of
sound. Note that there are plenty of general-aviation aircraft
(including some piston-driven aircraft and practically all jets)
that go fast enough to make the first-order approximation
unacceptable. This explains why we distinguish CAS versus EAS –
that is, calibrated airspeed versus equivalent airspeed. (This is
sometimes called the “compressibility correction” but that is a
terrible misnomer, as discussed in the next item.)

Beware that people sometimes claim that Bernoulli’s
principle only applies to “incompressible” fluids, but this claim
is nonsense. There are no incompressible fluids. Air is highly
compressible. The density ρ changes when the pressure P
changes. The temperature changes too. All these changes are quite
significant. Equations such as equation 3.5
already account for this correctly to first order; if they did
not, the equations would give spectacularly wrong answers. The
underlying idea is that since all these contributions are
proportional to one another (to first order), you don’t need a
temperature-dependent term and a density-dependent term and a
pressure-dependent term; you can lump all the dependencies into a
single term, which shows up as the pressure-dependent term in
equation 3.5. For details on this, see
section 3.6.3.

There are no incompressible fluids.

Besides, we are free to analyze things in a way that is correct to
second order (as in equation 3.8) or even
correct to all orders (as in equation 3.7); we are not
limited to the first-order approximations that are embodied in
equation 3.5.

These equations apply only to steady flow. That is, the
pressure, velocity, density, etc. can change from place to place, but
they must not change as a function of time at any particular place
(in your frame of reference).

For example, these equations do not apply inside
a cylinder with a piston, such as you find in a bicycle pump, or in
the cylinders of a piston engine. In such a cylinder, at any
particular location, the pressure changes as a function of time.
(It is possible to analyze such devices, but it requires formulas
that are more complicated than equation 3.5.)

Here’s another important example: Consider a vortex (such as a
tornado or hurricane) spinning counterclockwise and embedded in an
airmass that is moving northward relative to the ground. At a point
ahead of the vortex and to the left of the centerline, the pressure
will drop and the windspeed will drop as the vortex approaches.
This is just the opposite of what you would expect if you rashly
tried to apply equation 3.5 to this situation.

In this scenario, it would be valid to apply Bernoulli’s principle
in a frame moving along with the overall air mass, but not in a
frame attached to the ground.

So, we see that even when we do have steady flow, it will only be
steady in one frame of reference; it will be time-varying in almost
any other frame. This is one of the reasons why we usually choose
to analyze things using a frame moving along with the airplane.

The equations given here apply only to a particular parcel of
fluid moving along a particular streamline. The same logic
applies to every other parcel of fluid, but different parcels
will in general have different stagnation pressures.

Since we are restricting attention to steady
flow, any parcel anywhere on the same streamline will have the same
stagnation pressure. That’s because they are all essentially earlier
or later versions of the same parcel.

In some cases,8 we may have reason to
believe that every parcel in some vicinity started out with the
same stagnation pressure. This is not something that comes out of
the Bernoulli equation, but rather something that can be put into
the Bernoulli equation if/when we know it to be true.

Otherwise you should not use equation 3.5
to compare one parcel with another.

None of the equations given here be trusted in any situation
where frictional forces are playing a significant role. In
particular, in the “boundary layer” very near the surface
of a wing, there is a tremendous amount of friction, due to the
large difference in velocity between nearby points.

This proviso is not as much trouble as you might think, for the
following reasons. (1) In normal flight (not near the stall) the
boundary layer is usually very thin, and (2) we can apply
Bernoulli’s principle outside the boundary layer and
then infer by other means what the pressure must be inside
the boundary layer.

In many cases (but not all) the pressure inside the boundary
layer is very nearly the same as the pressure just outside
the boundary layer ... but you cannot use Bernoulli’s principle
to establish this fact. If you completely ignored the
existence of a boundary layer and tried to use Bernoulli’s
formula right near the surface of the wing, you would be
making two mistakes: using the wrong formula and using the
wrong airspeed. Oddly enough, in many cases these
two mistakes cancel each other out, but you should
not make a habit of doing things this way.

Bernoulli’s principle is intimately related to the idea of streamline
curvature discussed in section 3.5. If the
parcel experiences a side-to-side pressure gradient, the direction of
motion will change. If the parcel experiences a front-to-back
pressure gradient, the speed of motion will change. This is exactly
what we would expect from Newton’s laws of motion.

It must be emphasized that you do not get to choose Bernoulli’s
principle “instead of” Newton’s laws or vice versa. Bernoulli’s
principle is a consequence of Newton’s laws. See
section 3.16 for more on this.

Sometimes people who use equation 3.5 are tempted to
interpret it as an application of the principle of conservation of
energy. That is, they try to interpret Bernoulli’s equation as
equivalent to the law of the roller coaster (figure 1.9)
in the sense that the parcel loses speed when it climbs up a pressure
gradient and gains speed when it slides down a pressure gradient.
This is plausible at the level of dimensional analysis, since
½ ρ v2 is in fact the kinetic energy per unit volume, and
pressure has the same dimensions as energy per unit volume. Alas,
this interpretation is not correct. There is more to physics than
dimensional analysis. The parcel of air is unlike a roller coaster in
that it changes size and shape as it flows up and down the pressure
gradient. Furthermore, the pressure is not numerically equal to
the potential energy per unit volume. Actually, for nonmoving air,
the pressure is numerically equal to about 40% of the energy per unit
volume.

It is correct to interpret equation 3.5
as saying the enthalpy of the parcel remains constant.

It is not correct to say that the energy of the parcel
remains constant. There
is typically a qualitative correlation between the
enthalpy and the energy, in the sense that they both go up together
or both go down together … but they are not numerically equal.

It makes sense to measure the local velocity (lower-case
v) at each point as a multiple of the free-stream velocity
(capital V) since they vary in proportion to each other.
Similarly it makes sense to measure relative pressures in terms
of the free-stream dynamic pressure:

which is always small compared to atmospheric pressure (assuming
V is small compared to the speed of sound). Remember, this Q (with a
capital Q) is a property of the free stream, as measured far from
the wing.

Turning now to the local velocity v (with a small v)
and other details of the local flow pattern, the pressure versus
velocity relationship is shown graphically in figure 3.10.
The highest possible pressure (corresponding to completely stopped
air) is one Q above atmospheric, while fast-moving air can have
pressure several Q below atmospheric.

It doesn’t matter whether we measure P as an absolute pressure
or as a relative pressure (relative to atmospheric). If you change
from absolute to relative pressure it just shifts both sides of
Bernoulli’s equation by a constant, and the new value (just as before)
remains constant as the air parcel flows past the wing. Similarly, if
we use relative pressure in figure 3.10, we can drop the word
“Atm” from the pressure axis and just speak of “positive one Q”
and “negative two Q” — keeping in mind that all the pressures
are only slightly above or below one atmosphere.

Bernoulli’s principle allows us to understand why there is a positive
pressure bubble right at the trailing edge of the wing (which is the
last place you would expect if you thought of the air as a bunch of
bullets). The air at the stagnation line is the slowest-moving air in
the whole system; it is not moving at all. It has the highest
possible pressure, namely 1 Q.

As we saw in the bottom panel of figure 3.9, at high angles of
attack a wing is extremely effective at speeding up the air above the
wing and retarding the air below the wing. The maximum local velocity
above the wing can be more than twice the free-stream velocity.
This creates a negative pressure (suction) of more than 3 Q.

The
airplane’s altimeter operates by measuring the pressure at the
static port. See section 20.2.2 for more on
this.

The static port is oriented sideways to the airflow, at a point
where the air flows past with a local velocity just equal to the
free-stream velocity.

In accordance with Bernoulli’s principle, this velocity must be
associated with a “lower” pressure there.

You might think this lower pressure would cause huge errors in
the altimeter, depending on airspeed. In fact, though, there are no
such errors. The question is, why not?

The answer has to do with the notion of “lower” pressure. You have
to ask, lower than what? Indeed the pressure there is 1 Q lower than the stagnation pressure
of the air. However, in your reference
frame, the stagnation pressure is
1 Atm + 1
Q. When we subtract 1
Q from that, we see that the pressure in the static port is
just equal to atmospheric. Therefore the altimeter gets the right
answer, independent of airspeed.

Another way of saying it is that the air near the static port has 1
Atm of static pressure and 1 Q of dynamic pressure. The altimeter
is sensitive only to static pressure, so it reads 1 Atm — as it
should.

In contrast, the air in the Pitot tube has the same stagnation
pressure, 1 Atm + 1
Q, but it is all in the form of pressure since (in your
reference frame) it is not moving.

We can now see why the constant on the right hand side of
equation 3.5 is officially called the
stagnation pressure, since it is the pressure that you observe
in the Pitot tube or any other place where the air is stagnant,
i.e. where the local velocity v is zero (relative to the airplane).

In ordinary language “static” and “stagnant” mean almost the same
thing, but in aerodynamics they designate two very different concepts.
The static pressure is the pressure you would measure in the
reference frame of the air, for instance if you were in a balloon
comoving with the free stream. As you increase your airspeed, the
stagnation pressure goes up, but the static pressure does not.

Also: we can contrast this with what happens in a carburetor.
There is no change of reference frames, so the stagnation pressure remains
1 Atm. The high-speed air in the throat of the Venturi has a pressure
below the ambient atmospheric pressure.

Non-experts may not make much distinction between a “pressurized”
fluid and a “compressed” fluid, but in the engineering literature
there is a world of difference between the two concepts.

Every substance on earth is compressible — be it air, water, cast
iron, or anything else. It must increase its density when you apply
pressure; otherwise there would be no way to balance the energy
equations.

However, changes in density are not very important to understanding
the basic features of how wings work, as long as the airspeed is not
near or above the speed of sound. Typical general aviation airspeeds
correspond to Mach 0.2 or 0.3 or thereabouts (even when we account for
the fact that the wing speeds up the air locally), and at those speeds
the density never changes more than a few percent.

For an ideal gas such as air, density is proportional to pressure, so
you may be wondering why pressure-changes are important but
density-changes are not. Here’s why:

We are directly interested in differences in pressure.

We
are only rarely interested in differences in density.

We are only rarely interested in the total pressure.

We are
directly interested in the total density.

That is, lift depends on a pressure difference between the top
and bottom of the wing. Similarly pressure drag depends on pressure
differences. Therefore the relevant differential pressures are
zero plus important terms proportional to ½ρV2. Meanwhile, the
relevant pressures are proportional to the total density, which is
some big number plus or minus unimportant terms proportional to
½ρV2.

To say it again: Flight depends directly on total density but
not directly on total atmospheric pressure, just differences in
pressure.

Many books say the air is “incompressible” in the subsonic
regime. That’s bizarrely misleading. In fact, when those books use
the words “incompressible flow” it generally means that the density
undergoes only small-percentage changes. This has got nothing to do
with whether the fluid has a high or low compressibility. The real
explanation is that the density-changes are small because the
pressure-changes are small compared to the total atmospheric pressure.

As previously mentioned, many books claim that equation 3.4
only applies to an “incompressible” fluid,
but this claim is nonsense. Here’s the real story:

Compressibility specifies to first order how density depends on
pressure. Equation 3.4 specifies to first order how the
kinetic energy depends on pressure. It already accounts for the effects of
compressibility and all other first-order quantities. Therefore
equation 3.4 is valid whenever the pressure-changes are a small
percentage of the total pressure, regardless of compressibility.

At high airspeeds, the pressure changes are bigger, and you need
a more sophisticated form of Bernoulli’s equation.
Using the full equation of state, you can derive Bernoulli’s
equation in a form that is valid over the whole range of pressures
and speeds.

Here H/m is the specific enthalpy, i.e. the enthalpy per unit mass,
as explained in reference 24.
Also, P0 is some “reference” pressure (usually taken to be the
ambient atmospheric pressure), ΔP is the difference between
the actual pressure and the reference pressure, ρ0 is the
density the air would have at the reference pressure, and γ (gamma)
is a constant that appears in the equation of state for the fluid. It
is sometimes called the adiabatic exponent, and sometimes called the
ratio of specific heats, for reasons that need not concern us at the
moment. The γ value for a few fluids are given in the table
below.

The validity of the approximations involved in equation 3.4 do not depend on any notion of “incompressible”
fluid, as we can see from the fact that the correction term depends on
γ, which is not correlated with the actual compressibility.

dimensionless

adiabatic

γ

compressibility

helium

1.666

0.6

nitrogen

1.4

0.714

oxygen

1.4

0.714

air

1.4

0.714

methane

1.31

0.763

The meaning of the numbers in the rightmost column in the table is
this: If you start with a sample of air and increase the pressure by
1%, the volume goes down by 0.7%.

In equation 3.8, when the pressure P is near
atmospheric, the term in square brackets approaches unity, and the
expression becomes equivalent to the elementary version,
equation 3.4, as it should.

Don’t let anybody tell you that Bernoulli’s principle can’t cope with
compressibility. Even the elementary version (equation 3.4)
accounts for compressibility to first order.

We are now in a position to understand how stall warning devices
work. There are two types of stall-warning devices commonly used on
light aircraft. The first type (used on most Pipers, Mooneys, and
Beechcraft) uses a small vane mounted slightly below and aft of the
leading edge of the wing as shown in the left panel of
figure 3.11. The warning is actuated when the vane is blown
up
and forward. At low angles of attack (e.g. cruise) the stagnation
line is forward of the vane, so the vane gets blown backward and
everybody is happy. As the angle of attack increases, the stagnation
line moves farther and farther aft underneath
the wing. When it has moved farther aft than the vane, the air will
blow the vane forward and upward and the stall warning will be
activated.

The second type of stall-warning device (used on the Cessna 152, 172,
and some others, not including the 182) operates on a different
principle. It is sensitive to suction at the surface rather than flow
along the surface. It is positioned just below the leading edge of
the wing, as indicated in the right panel of figure 3.11.
At low angles of attack, the leading edge is a low-velocity,
high-pressure region; at high angles of attack it becomes a
high-velocity, low-pressure region. When the low-pressure region
extends far enough down around the leading edge, it will suck air out
of the opening. The air flows through a harmonica reed, producing an
audible warning.

Note that neither device actually detects the stall. Each one really
just measures angle of attack. It is designed to give you a warning a
few degrees before the wing reaches the angle of attack where
the stall is expected. Of course if there is something wrong, such as
frost on the wings (see section 3.15),
the stall will occur at a lower-than-expected angle of attack, and you
will get no warning from the so-called stall warning device.

We all know that at the submicroscopic level, air consists of
particles, namely molecules of nitrogen, oxygen, and
various other substances. Starting from the properties of these
molecules and their interactions, it is possible to calculate
macroscopic properties such as pressure, velocity, viscosity,
speed of sound, et cetera.

However, for ordinary purposes such as understanding how wings work,
you can pretty much forget about the individual particles, since the
relevant information is well summarized by the macroscopic properties
of the fluid. This is called the hydrodynamic
approximation.

In fact, when people try to think about the individual
particles, it is a common mistake to overestimate the size of
the particles and to underestimate the importance of the interactions
between particles.

If you erroneously imagine that air particles are large and
non-interacting, perhaps like the bullets shown
in figure 3.12, you will never understand how wings work.
Consider the following comparisons. There is only one important thing
bullets and air molecules have in common:

Bullets hit the bottom of the wing, transferring upward momentum
to it.

Similarly, air molecules hit the bottom of the wing, transferring
upward momentum to it.

Otherwise, all the important parts of the story are different:

No bullets hit the top of the wing.

Air pressure on top of the wing is only a few percent lower than the
pressure on the bottom.

The shape of the top of the wing doesn’t matter to the bullets.

The shape of the top of the wing is crucial. A spoiler at
location “X” in figure 3.12 could easily double the drag of
the entire airplane.

The bullets don’t hit each other, and
even if they did, it wouldn’t affect lift production.

Each air molecule collides with one or another of its neighbors
10,000,000,000 times per second. This is crucial.

Each bullet weighs a few grams.

Each nitrogen molecule weighs
0.00000000000000000000005 grams.

Bullets that miss the wing are undeflected.

The wing creates a pressure field that strongly deflects
even far-away bits of fluid, out to a distance of a wingspan or so in
every direction.

Bullets could not possibly knock a stall-warning vane forward.

Fluid flow nicely explains how such a vane gets blown
forward and upward. See section 3.7.

The list goes on and on, but you get the idea. Interactions between
air molecules are a big part of the story. It is a much better
approximation to think of the air as a continuous fluid
than as a bunch of bullet-like particles.

You may have heard stories that try to use the Coandǎ effect or
the teaspoon effect to explain how wings produce lift. These
stories are completely fallacious, as discussed in
section 18.4.4 and section 18.4.3.

There are dozens of other fallacies besides. It is beyond the scope
of this book to discuss them, or even to catalog them all.

You’ve probably been told that an airfoil produces lift because it is
curved on top and flat on the bottom. This is “common
knowledge” ... but alas it’s not true. You shouldn’t believe it, not
even for an instant.

Presumably you are aware that airshow pilots routinely fly for
extended periods of time upside down. Doesn’t
that make you suspicious that there might be something wrong with the
story about curved on top and flat on the bottom?

Here are some of things you expect to see in an airplane optimized for
upside-down flight:

Four-point or
five-point seatbelts, to keep the pilot from flopping around.

Flop tubes in the fuel tanks, so that the engine
continues to run during extended inverted flight.

Possibly a stronger structure, to handle stresses in
funny directions.

Possibly additional windows, for looking out in funny
directions.

Possibly a less-cambered wing, to optimize the inverted stall,
but not necessarily. A C-150 Aerobat has exactly the same wing as a
regular C-150.

Possibly less washout, again to optimize the inverted stall,
but again not necessarily.

Fuel injection (as opposed to carburetion), although this is
hardly worth mentioning, because nowadays practically everything
bigger than a walk-behind lawn mower is fuel injected, and even some
lawn mowers.

et cetera.

It must be emphasized that changing the cross-sectional shape of the
wing is not a necessity. Any ordinary wing flies just fine inverted.
Even a wing that is flat on one side and curved on the other flies
just fine inverted, as shown in figure 3.13. It may look a bit
peculiar, and the inverted stall is not optimized, but basically it
just works. The basic lift-producing process is the same.

The misconception that wings must be curved on top and flat on the
bottom is commonly associated with the previously-discussed
misconception that the air is required to pass above and below the
wing in equal amounts of time. In fact, an upside-down wing produces
lift by exactly the same principle as a rightside-up wing.

To help us discuss airfoil shapes, figure 3.14
illustrates some useful terminology.

The chord line is the straight line drawn from the
leading edge to the trailing edge.

The term camber in
general means “bend”. If you want to quantify the amount of
camber, draw a curved line from the leading edge to the trailing edge,
staying always halfway between the upper surface and the lower
surface; this is called the mean camber line. The maximum
difference between this and the chord line is the amount of camber.
It can be expressed as a distance or (more commonly) as a percentage
of the chord length.

A symmetric airfoil, where the top surface is a mirror image of
the bottom surface, has zero camber. The airflow and pressure
patterns for such an airfoil are shown in figure 3.15.

This figure could be considered the side view of
a symmetric wing, or the top view of a rudder. Rudders are airfoils,
too, and work by the same principles.

At small angles of attack, a symmetric airfoil works
better than a highly cambered airfoil. Conversely, at high angles
of attack, a cambered airfoil works better than the corresponding
symmetric airfoil. An example of this is shown in figure 3.16.
The airfoil designated “631-012” is symmetric, while
the airfoil designated “631-412” airfoil is cambered;
otherwise the two are pretty much the same.9 At any normal angle of attack (up to
about 12 degrees), the two airfoils produce virtually identical
amounts of lift. Beyond that point the cambered airfoil has a
big advantage because it does not stall until a much higher relative
angle of attack. As a consequence, its maximum coefficient of
lift is much greater.

At high angles of attack, the leading edge of a cambered
wing will slice into the wind at less of an angle compared to
the corresponding symmetric wing. This doesn’t prove anything,
but it provides an intuitive feeling for why the cambered wing
has more resistance to stalling.

On some airplanes, the airfoils have no camber at all, and on most of
the rest the camber is barely perceptible (maybe 1 or 2 percent). One
reason wings are not more cambered is that any increase would require
the bottom surface to be concave — which would be a pain to
manufacture. A more profound reason is that large camber is only really
beneficial near the stall, and it suffices to create lots of camber by
extending the flaps when needed, i.e. for takeoff and landing.

Reverse camber is clearly a bad idea (since it causes
earlier stall) so aircraft that are expected to perform well upside
down (e.g. Pitts or Decathlon) have symmetric (zero-camber) airfoils.

We have seen that under ordinary conditions, the
amount of lift produced by a wing depends on the angle of attack,
but hardly depends at all on the amount of camber. This makes
sense. In fact, the airplane would be unflyable if the coefficient
of lift were determined solely by the shape of the wing. Since
the amount of camber doesn’t often change in flight, there would
be no way to change the coefficient of lift. The airplane could
only support its weight at one special airspeed, and would be
unstable and uncontrollable. In reality, the pilot (and the trim
system) continually regulate the amount of lift by regulating
the all-important angle of attack; see chapter 2 and
chapter 6.

It is thin, highly cambered, and quite concave on the bottom. There
is no significant difference between the top surface and the bottom
surface — same length, same curvature. Still, the wing produces
lift, using the same lift-producing principle as any other airfoil.
This should further dispel the notion that wings produce lift because
of a difference in length between the upper and lower surfaces.

Similar remarks apply to the sail of a sailboat. It is a very thin
wing, oriented more-or-less vertically, producing sideways lift.

Even a thin flat object such as a barn door will produce lift,
if the wind strikes it at an appropriate angle of attack. The airflow
pattern (somewhat idealized) for a barn door (or the wing on a
dime-store balsa glider) is shown in figure 3.18. Once again,
the lift-producing mechanism is the same.

You may be wondering whether the flow patterns shown in
figure 3.18 or the earlier figures are the only ones allowed by
the laws of hydrodynamics. The answer is: almost, but not quite.
Figure 3.19 shows the barn door operating with the same angle of
attack (and the same airspeed) as in figure 3.18, but the airflow
pattern is different.

The new airflow pattern (figure 3.19) is
highly symmetric. I have deleted the timing information, to make
it clear that the stream lines are unchanged if you flip the figure
right/left and top/bottom. The front stagnation line is a certain
distance behind the leading edge; the rear stagnation line is
the same distance ahead of the trailing edge. This airflow pattern
produces no lift. (There will be a lot of torque — the so-called Rayleigh torque — but no
lift.)

To understand circulation and its effects, first imagine an airplane
with barn-door wings, parked on the ramp on a day with no wind. Then
imagine stirring the air with a paddle, setting up a circulatory flow
pattern, flowing nose-to-tail over the top of the wing and
tail-to-nose under the bottom (clockwise in this figure). This is the
flow pattern for pure circulation, as shown in figure 3.20.
The magnitude of this circulatory flow is greatest near the wing,
and is negligible far from the wing. It does not affect the airmass
as a whole.

Then imagine that a headwind springs up, a steady overall wind blowing
in the nose-to-tail direction (left to right in the figure), giving
the parked airplane some true airspeed relative to the airmass as a
whole. At each point in space, the velocity fields will add. The
circulatory flow and the airmass flow will add above the wing,
producing high velocity and low pressure there. The circulatory flow
will partially cancel the airmass flow below the wing, producing
low velocity and high pressure there.

If we take the noncirculatory nose-to-tail flow in figure 3.19
and add various amounts of circulation, we can generate all the
flow patterns consistent with the laws of hydrodynamics — including
the actual natural airflow shown in figure 3.18 and
figure 3.21.10

If you suddenly accelerate a wing from a standing
start, the initial airflow pattern will be noncirculatory, as
shown in figure 3.22. Fortunately for us, the air
absolutely hates this airflow pattern, and by the time the wing
has traveled a short distance (a couple of chord-lengths or so)
it develops enough circulation to produce the normal airflow
pattern shown in figure 3.24.

In real flight situations, precisely enough circulation will be
established so that the rear stagnation line is right at the trailing
edge, so no air needs to turn the corner there. The counterclockwise
flow at the trailing edge in figure 3.19 is cancelled by the
clockwise flow in figure 3.20. Meanwhile, at the leading edge,
both figure 3.19 and figure 3.20 contribute clockwise
flow, so the real flow pattern (figure 3.21) has lots and lots
of flow around the leading edge.

The general rule — called the Kutta condition — is that the
air hates to turn the corner at a sharp trailing edge. To a first
approxmation, the air hates to turn the corner at any sharp
edge, because the high velocity there creates a lot of friction.
For ordinary wings, that’s all we need to know, because the trailing
edge is the only sharp edge.

The funny thing is that if the trailing edge is sharp, an airfoil will
work even if the leading edge is sharp, too. This explains why
dime-store balsa-wood gliders work, even with sharp leading edges.

It is a bit of a mystery
why the air hates turning a corner at the trailing edge, and doesn’t
mind so much turning a sharp corner at the leading edge — but
that’s the way it is.11 This is related to the well-known fact
that blowing is different from sucking. (Even though you can blow out
a candle from more than a foot away, you cannot suck out a candle from
more than an inch or two away.) In any case, the rule is:

The air wants to flow cleanly off the trailing edge.

As the angle of attack increases, the amount of circulation needed
to meet the Kutta condition increases.

Here is a nice, direct way of demonstrating the Kutta condition:

Choose an airplane where the stall warning indicator is on the flapped
section of the wing. This includes the Cessna C-152 and
C-172, but not the C-182. It includes most Mooneys and the Grumman
Tiger, but excludes Piper Cherokees and the Beech Bonanza.

At a safe altitude, start with the airplane in the clean configuration
in level flight, a couple of knots above the speed where the stall
warning horn comes on.

The following items are not what we are trying
to emphasize here, but for completeness they should perhaps be
mentioned: (a) since extending the flaps increases the coefficient
of lift the wing can produce, you can expect to need a lower
airspeed, in order to maintain lift equal to weight; (b) you may
need to fiddle with the throttle in order to maintain level
flight; and (c) you may need to fiddle with the yoke to keep the
fuselage at a constant pitch angle.

The goal is to create a situation where increasing the incidence of
the wing section – by extending the flaps – increases the section’s
angle of attack and increases its circulation. The increased
circulation trips the stall-warning detector, as described in
section 3.7.

We need to maintain the fuselage at a constant angle relative to the
direction of flight, so that changing the incidence directly changes
the wing’s angle of attack, in accordance with the formula pitch +
incidence = angle of climb + angle of attack, as discussed in
section 2.4.

There is no need to stall the airplane; the warning horn itself makes
the point.

This demonstration makes it clear that the flap (which is at the
back of the wing) is having a big effect on the airflow around
the entire wing, including the stall-warning detector (which is
near the front).

Here is a beautifully simple and powerful result: The lift is equal to
the airspeed, times the circulation, times the density of the air,
times the span of the wing. This is called the Kutta-Zhukovsky
theorem.12

Since circulation is proportional to the coefficient of lift and
to the airspeed, this new notion is consistent with our previous
knowledge that the lift should be proportional to the coefficient
of lift times airspeed squared.

You can look at a velocity field and visualize the circulation. In
figure 3.25, the right-hand edge of the blue streamers shows
where the air is 70 milliseconds after passing the reference point.
For comparison, the vertical black line shows where the 70 millisecond
timeline would have been if the wing had been completely absent.
However, this comparison is not important; you should be comparing
each air parcel above the wing with the corresponding parcel below the
wing.

Because of the circulatory contribution to the velocity, the streamers
above the wing are at a relatively advanced position, while the
streamers below the wing are at a relatively retarded position.

If you refer back to figure 3.9, you can see that circulation
is proportional to angle of attack. In particular, note that
when the airfoil is not producing lift there is no circulation
— the upper streamers are not advanced relative to the lower streamers.

The same thing can be seen by comparing figure 3.22
to figure 3.24 — when there is no circulation the upper
streamers are not advanced relative to the lower streamers.

Circulation can be measured, according to the following procedure.
Set up an imaginary loop around the wing. Go around the loop
clockwise, dividing it into a large number of small segments. For
each segment, multiply the length of that segment times the speed of
the air along the direction of the loop at that point. (If the
airflow direction is opposite to the direction of the loop, the
product will be negative.) Add up all the products. The total
velocity-times-length will be the circulation. This is the
official definition.

Interestingly, the answer is essentially independent of the size and
shape of the loop.13 For instance, if you go farther
away, the velocity will be lower but the loop will be longer, so the
velocity-times-length will be unchanged.

There is a widely-held misconception that it is the
velocity relative to the skin of the wing that produces
lift. This causes no end of confusion.

Remember that the air has a well defined velocity and pressure
everywhere, not just at the surface of the wing. Using a windmill and
a pressure gauge, you can measure the velocity and pressure anywhere
in the air, near the wing or elsewhere. The circulatory flow set up
by the wing creates low pressure in a huge region extending far above
the wing. The velocity at each point determines the pressure at that
point.

The circulation near a wing is normally set up by the interaction of
the wind with the shape of the wing. However, there are other ways of
setting up circulatory flow. In figure 3.26, the wings are not
airfoil-shaped but paddle-shaped. By rotating the paddle-wings, we can set up a circulatory
airflow pattern by brute force.

Bernoulli’s principle applies point-by-point in the air near the wing,
creating low pressure that pulls up on the wings, even though the air
near the wing has no velocity relative to the wing – it is “stuck”
between the vanes of the paddle. The Kutta-Zhukovsky theorem remains
the same as stated above: lift is equal to the airspeed, times the
circulation, times the density of the air, times the span of the wing.

This phenomenon — creating the circulation needed for lift by
mechanically stirring the air — is called the Magnus effect.

The airplane in figure 3.26 would have definite
controllability problems, since the notion of angle of attack
would not exist (see chapter 2 and chapter 6).
The concept, though, is not as ridiculous as might seem. The
famous aerodynamicist Flettner once built a ship that “sailed” all the way across the Atlantic
using huge rotating cylinders as “sails” to catch the wind.

Also, it is easier
than you might think to demonstrate this important concept. You don’t
need four vanes on the rotating paddle; a single flat surface will do.
A business card works fairly well. Drop the card from shoulder
height, with its long axis horizontal. As you release it, give it a
little bit of backspin around the long axis. It will fly surprisingly
well; the lift-to-drag ratio is not enormous, but it is not zero
either. The motion is depicted in figure 3.27.

You can improve the performance by giving the wing
a finer aspect ratio (more span and/or less chord). I once took
a manila folder and cut out several pieces an inch wide and 11
inches long; they work great.

As an experiment, try giving the wing the wrong direction
of circulation (i.e. topspin) as you release it. What do you
think will happen?

I strongly urge you to try this demonstration yourself.
It will improve your intuition about the relationship of circulation
and lift.

We can use these ideas to understand some (but not all) of the
aerodynamics of tennis balls and similar
objects. As portrayed in figure 3.28, if a ball is hit with a lot
of backspin, the surface of the spinning ball will create the
circulatory
flow pattern necessary to produce lift, and it will be a “floater”.
Conversely, the classic “smash” involves topspin, which produces
negative lift, causing the ball to “fly” into the ground faster than
it would under the influence of gravity alone. Similar words apply to
leftward and rightward curve balls.

To get even close to the right answer, we must ask where the relative
wind is fast or slow, relative to undisturbed parcels of air — not
relative to the rotating surface of the ball. Remember that the fluid
has a velocity and a pressure everywhere, not just at the surface of
the ball. Air moving past a surface creates drag, not
lift. Bernoulli says that high velocity is associated with
low pressure and vice versa.
For the floater, the circulatory flow created by the
backspin combines with the free-stream flow created by the ball’s
forward motion to create high-velocity, low-pressure air above the
ball — that is, lift.

The air has velocity and pressure
everywhere ... not just at surfaces.

This simple picture of mechanically-induced circulation
applies best to balls that have evenly-distributed roughness.
Cricket balls are in a different category, since they have a
prominent equatorial seam. If you spin-stabilize the orientation
of the seam, and fly the seam at an “angle of attack”,
airflow over the seam causes extra turbulence which promotes attached
flow on one side of the ball. See section 18.3
for some discussion of attached versus separated flow. Such effects
can overwhelm the mechanically-induced circulation.

To really understand flying balls or cylinders, you
would need to account for the direct effect of spin on circulation,
the effect of spin on separation, the effect of seams on separation,
et cetera. That would go beyond the scope of this book. A wing
is actually easier to understand.

A vortex is a bunch of air circulating around itself. The axis
around which the air is rotating is called a vortex line. It is
mathematically impossible for a vortex line
to have loose ends. A smoke ring is an example of a vortex. It
closes on itself so it has no loose ends.

The circulation necessary to produce lift can be attributed to a
bound vortex line. It binds to the wing and travels with the
airplane. The question arises, what happens to this vortex line at
the wingtips?

In the simplest case, the answer is that the vortex spills off each
wingtip. Each wing forms a trailing vortex (also called wake vortex) that extends for
miles behind the airplane. These trailing vortices constitute the
continuation of the bound vortex. See figure 3.29. Far behind
the airplane, possibly all the way back at the place where the plane
left ground effect, the two trailing vortices join up to form an
unbroken14 vortex line.

The air rotates around the vortex line in the direction
indicated in the figure. We know that the airplane, in order
to support its weight, has to yank down on the air. The air that
has been visited by the airplane will have a descending motion
relative to the rest of the air. The trailing vortices mark the
boundary of this region of descending air.

It doesn’t matter
whether you consider the vorticity to be the cause or the effect
of the descending air — you can’t have one without the other.

Lift must equal weight times load factor, and we can’t easily change
the weight, or the air density, or the wingspan. Therefore, when the
airplane flies at a low airspeed, it must generate lots of
circulation.

It is a common misconception that the wingtip vortices are somehow
associated with unnecessary spanwise flow (sometimes called
“lateral” flow), and that they can be eliminated using fences,
winglets, et cetera. The reality is that the vortices are
completely necessary; you cannot produce lift without producing
vortices.

Lift and trailing vortices are intimately and necessarily
associated with air flowing around the span.

Neither lift nor trailing
vortices are in any important way associated with “lateral” flow
along the span.

Also keep in mind that “circulation” and “vorticity” are two quite
different ways of expressing the same idea: When we draw a vortex
line, it represents the core of the vortex, which is the axis of the
circulatory motion. The air circulates around the vortex line.
Circulation refers to flow around the vortex line, not along the
vortex line.

If you look closely, you find that the overall flow pattern is more
accurately described by a large number of weak vortex lines, rather
than by the one strong vortex line shown in figure 3.29.
By fiddling with the shape of the wing the designers can control
(to some extent) where along the span the vorticity is shed.

It turns out that behind each wing, the weak vortex lines get twisted
around each other. (This is the natural consequence of the fact that
each vortex line gets carried along in the circulatory flow of each of
the other vortex lines.) If you look at a point a few span-lengths
behind the aircraft, all the weak vortex lines have rolled up into
what is effectively one strong vortex. That means that visualizing
the wake in terms of one strong vortex (per wingtip), as shown in
figure 3.29, is good enough for most pilot purposes. However,
you might care about the details of the roll-up process if you are
flying in close formation behind another aircraft, such as a glider
being towed.

Winglets encourage the vorticity to be shed nearer the wingtips,
rather than somewhere else along the span. This produces more lift,
since each part of the span contributes lift in proportion to the
amount of circulation carried by that part of the span, in
accordance with the Kutta-Zhukovsky theorem. In any case, as a
general rule, adding a pair of six-foot-tall winglets has no
aerodynamic advantage compared to adding six feet of regular,
horizontal wing on each side.15

The important point remains that there is no way to produce lift
without producing wake vortices. Remember: The trailing vortices
mark the boundary between the descending air behind the wing and the
undisturbed air outboard of the descending region.

The bound vortex that produces the circulation that
supports the weight of the airplane should not be confused with
the little vortices produced by vortex generators (to re-energize
the boundary layer) as discussed in section 18.3.

When air traffic control (ATC) tells you “caution — wake
turbulence” they are really telling you that some previous airplane
has left a wake vortex in your path. The wake
vortex from a large, heavy aircraft can easily flip a small aircraft
upside down.

A heavy airplane like a C5-A flying slowly is the biggest threat,
because it needs lots of circulation to support all that weight at a
low airspeed. So the most important rule is to beware of an aircraft
that is heavy and slow.

Conventional pilot lore says that an aircraft with flaps extended
should be less dangerous than one with the flaps retracted, on
the grounds that there is more circulation around the flapped section
of wing, and less circulation around the remaining (outboard) section
of each wing. That means that a goodly amount of circulation will be
shed at the boundary between the flapped and unflapped section, so you
get two half-strength vortices per wing, rather than one full-strength
one.

That’s undoubtedly relevant if you are flying in close formation
behind a heavy, slow aircraft … but in the other 99.999% of
general-aviation flying, you won’t be close enough for the other
plane’s flaps to give you any protection. At any reasonable distance
behind the other aircraft, all the trailing vorticity will have rolled
up into what is effectively one strong vortex. When you couple that
with the fact that the aircraft with flaps extended might be flying
slower than the one without, you should not imagine that flaps reduce
the threat of wake turbulence. Besides, I don’t plan on getting close
enough to the other aircraft to even see whether it’s got flaps
extended or not.

To summarize: Although conventional pilot lore says to beware of heavy,
slow, and clean, it is simpler and better to beware of heavy and slow
(whether clean or not).

Beware of vortices behind heavy and slow aircraft.

Like a common smoke ring, the wake vortex does not
just sit there, it moves. In this case it moves downward. A
common rule of thumb says they normally descend at about 500 feet
per minute, but the actual rate will depend on the wingspan and
coefficient of lift of the airplane that produced the vortex.

Vortices are part of the air. A vortex in a moving airmass will be
carried along with the air. In fact, the reason wake vortices descend
is that the right vortex is carried downward by the flow field
associated with the left vortex, and the left vortex is carried
downward by the flow field associated with the right vortex.
Superimposed on this vertical motion, the ordinary wind blows
the vortices downwind, usually more-or-less horizontally.

When a vortex line gets close to the ground, it “sees its
reflection”. That is, a vortex at height H moves as if it were
being acted on by a mirror-image vortex a distance H below
ground. This causes wake vortices to spread out — the left vortex
starts moving to the left, and the right vortex starts moving to the
right.

If you are flying a light aircraft, avoid the airspace
below and behind a large aircraft. Avoiding the area for a minute
or two suffices, because a vortex that is older than that will
have lost enough intensity that it is probably not a serious problem.

If you are landing on the same runway as a preceding
large aircraft, you can avoid its wake vortices by flying a high,
steep approach, and landing at a point well beyond the point where
it landed. Remember, it doesn’t produce vortices unless it is
producing lift. Assuming you are landing into the wind, the wind
can only help clear out the vortices for you.

If you are departing from the same runway as a preceding large
aircraft, you can avoid its vortices — in theory — if you leave
the runway at a point well before the point where it did, and if
you make sure that your climb-out profile stays above and/or behind
its. In practice, this might be hard to do, since the other aircraft
might be able to climb more steeply than you can. Also, since you are
presumably taking off into the wind, you need to worry that the
wind might blow the other plane’s vortices toward you.

A light crosswind might keep a vortex on the runway
longer, by opposing its spreading motion. A less common problem is
that a crosswind might blow vortices from a parallel runway onto your
runway.

The technique that requires the least sophistication is to
delay your takeoff a few minutes, so the vortices can spread out and
be weakened by friction.

Here are some more benefits of understanding circulation
and vortices: it explains induced drag, and
explains why gliders have long skinny
wings. Induced drag is commonly said to be the “cost”
of producing lift. However, there is no law of physics that requires
a definite cost. If you could take a very large amount of air
and pull it downward very gently, you could support your weight
at very little cost. The cost you absolutely must pay is the
cost of making that trailing vortex. For every mile that the
airplane flies, each wingtip makes another mile of vortex. The
circulatory motion in that vortex involves nontrivial amounts
of kinetic energy, and that’s why you have induced drag. A long
skinny wing will need less circulation than a short fat wing producing
the same lift. Gliders (which need to fly slowly with minimum
drag) therefore have very long skinny wings (limited only by strength;
it’s hard to build something long, skinny, and strong).

We can use what we know about vortices to help understand
soft-field takeoff procedure.

By way of background, consider the following parable: Suppose you
throw a basketball downwards. By the law of action and reaction, this
gives you some upward momentum. Now the ball bounces, and you catch
it as it comes back up. The catch gives you some additional upward
momentum. The ball winds up with the same momentum before and after,
and the same energy before and after, but still you were able to use
it as a means to transfer some momentum.

The parable works like this:

Not in Ground Effect

In Ground Effect

Normal flight is like flinging a series of baskeballs
downward, using each one only once. Lift requires imparting some
momentum to each basketball, and this in turn requires some energy.

In ground effect, you get to catch the basketball as it comes back up
after bouncing. Ideally, this doubles the amount of useful momentum
transfer (a 100% increase) and approximately cancels the required
energy (a 100% decrease).

Let’s go through the analysis again:

In normal flight, producing lift requires transferring
momentum to the air. This momentum “eventually” gets transferred to
the ground, but that is a messy process that dissipates energy.

In
ground effect, the momentum gets rather quickly transferred
through the air into the ground. Ideally the air winds up with
the same momentum before and after, and the same energy before and
after.

You wind up with less momentum stored in the air,
but that’s not a problem, because lift depends on transferring
momentum through the air.

Let’s go through the analysis a third time, with yet more technical
detail:

When the aircraft is in ground effect, it “sees its
reflection” in the ground. If you are flying 10 feet above the
ground, you have a choice: (a) You can analyze it by saying the air
bounces off the ground, or (b) you can get rid of the ground and
instead imagine a mirror-image aircraft flying upside down, 10 feet
below where the ground level used to be. Instead of air bouncing
upward from the ground, you see upward-flowing air coming from the
mirror-image aircraft, as it produces upside-down lift.

Because you are flying in the upwash created by the mirror-image
airplane, you need a lower angle of attack to produce a given amount
of lift, at any given airspeed. You can think of this as a
pseudo-updraft. It is a relatively minor effect, comparable to the
pseudo-tailwind mentioned in section 3.14.5. It is most
noticeable in situations where it affects the tail differently from
the main wing, for instance when there is a pitch-change associated
with takeoff or with the landing flare. In such situations, the
airplane might exhibit nonlinear “squirrely” behavior.

It remains true that lift requires circulation and vortices, as
discussed in section 3.14. The circulatory
flow-pattern associated with your wing is still grabbing air and
throwing it downwards. This accounts for the force of lift supporting
your airplane.

Meanwhile, the mirror-image airplane is grabbing air and throwing it
upwards at you. Now, it turns out that in terms of force, a downward
flow of downward momentum is the same as an upward flow of
upward momentum. The forces don’t cancel. If you are very very near
the ground, you wind up with nearly 100% less momentum stored in the
air, and nearly 100% more momentum being transferred through the air.

The mirror-image aircraft’s trailing vortices spin in the opposite
direction and largely cancel your trailing vortices — greatly
reducing induced drag. The bound vortex attached to your wing is not
canceled. It is still there, as it must be to produce lift in
accordance with the Kutta-Zhukovsky theorem. However, after the
vortex spills off the wing it can mix with and cancel the mirror-image
trailing vortex. This is by far the most significant part of what we
call ground effect.

Here’s how this relates to pilot procedure: As discussed in section 13.4, in a soft-field takeoff, you leave the ground at a very
low airspeed, and then fly in ground effect for a while. There will
be no wheel friction (or damage) because the wheels are not touching
the ground. There will be very little induced drag because of the
ground effect, and there will be very little
parasite drag because you are going slowly. The
airplane will accelerate like crazy. When you reach normal flying
speed, you raise the nose and fly away.

Let’s not forget about the bound vortex, which runs
spanwise from wingtip to wingtip, as shown in figure 3.29.

When you are flying in ground effect, you are influenced by the mirror
image of your bound vortex. Specifically, the flow circulating around
the mirror-image bound vortex will reduce the airflow over your wing.
I call this a pseudo-tailwind.16
It is a relatively minor effect, comparable to the pseudo-updraft
mentioned in section 3.14.4.

Operationally, this means that for any given angle of attack, you need
a higher true airspeed to support the weight of the airplane. This in
turn means that a low-wing airplane will need a longer runway than the
corresponding high-wing airplane, other things being equal. It also
means – in theory – that there are tradeoffs involved during a
soft-field takeoff: you want to be sufficiently deep in ground effect
to reduce induced drag, but not so deep that your speeds are unduly
increased. In practice, though, feel free to fly as low as you want
during a soft-field takeoff, since in an ordinary-shaped airplane the
bad effect of the reflected bound vortex (greater speed) never
outweighs the good effect of the reflected trailing vortices (lesser
drag).

As a less-precise way of saying things, you could say that to
compensate for ground effect, at any given true airspeed, you need more
coefficient of lift. This explains why all airplanes – some more so
than others – exhibit “squirrely” behavior when flying near the
ground, including:

Immediately after liftoff, the airplane may seem to leap up a
few feet, as you climb out of the pseudo-tailwind. This is generally
a good thing, because when you become airborne you generally want to
stay airborne.

Conversely, on landing, the airplane may seem to drop suddenly,
as the pseudo-tailwind takes effect. In some slight, very theoretical
way this could be helpful, insofar as when you land you generally want
to stay landed, but in reality, you usually prefer the landing to be
more gentle rather than more firm, so it would be better to have
negative feedback, not positive feedback. In any case, it’s not
really a big problem once you learn to anticipate it. It does mean
that practicing flaring at altitude (as discussed in section 12.11.3) will never entirely prepare you for
real landings.

The wing and the tail will be influenced by ground effect to
different degrees. (This is particularly pronounced if your airplane
has a low wing and a high T-tail, but no airplane is entirely immune.)
That means that when you enter or exit ground effect, there will be
squirrely pitch-trim changes ... in addition to the effects mentioned
in the previous items. Just to rub salt in the wound, the behavior
will be different from flight to flight, depending on how the aircraft
is loaded, i.e. depending on whether the center of mass is near the
forward limit or the aft limit.

During landing, ground effect is a lose/lose/lose proposition. You
regret greater speed, you regret lesser drag, and you regret squirrely
handling.

The Federal Aviation Regulations prohibit takeoff when there is
frost adhering in critical places
including wings and control surfaces. The primary reason for concern
is that the frost causes roughness on the surface of the airfoil. (In
contrast, the weight of the frost is usually negligible.)

Wind-tunnel data indicates that roughness can cause a
surprisingly large amount of trouble.

The most obvious effect of roughness on the wings is to create
a lot more drag, as seen in the right panel in figure 3.30,
which shows wind-tunnel data for a real airfoil (the NACA 631-412
airfoil; see reference 26). At cruise angle of attack, the drag
is approximately doubled; at higher angles of attack (corresponding to
lower airspeeds) it is even worse.

The less obvious (yet more critical) problem is that
roughness causes the wing to stall at a considerably lower angle
of attack, lower coefficient of lift, and higher airspeed. This
can be seen in the left panel of figure 3.30. The pilot
of the frosty airplane could get a very nasty surprise.

As mentioned in section 3.6, Bernoulli’s
principle cannot be trusted when frictional forces are
at work. Frost, by sticking up into the breeze, is very
effective in causing friction. This tends to
de-energize the boundary layer, leading to separation which produces
the stall.17

It is interesting that at moderate and low angles
of attack (cruise airspeed and above) the frost has hardly any
effect on the coefficient of lift. This reinforces the point
made in section 3.13 that the velocity of the air right
at the surface, relative to the surface, is not what produces
the lift.

An interesting situation arises when the airplane has been sitting
long enough to pick up a big load of frost, but the present air
temperature is slightly above freezing. By far the easiest way
to get rid of the frost is by dousing the plane with five-gallon jugs
of warm water. That will melt the frost and heat the wings to an
above-freezing temperature (so that frost will not re-form).

The wing produces lift “because” of Newton’s law of action and
reaction.

We now examine the relationship between these physical
principles. Do we get a little bit of lift because of Bernoulli,
and a little bit more because of Newton? No, the laws of physics
are not cumulative in this way.

There is only one lift-producing process. Each of
the explanations itemized above concentrates on a different aspect
of this one process. The wing produces circulation in proportion
to its angle of attack (and its airspeed). This circulation means
the air above the wing is moving faster. This in turn produces
low pressure in accordance with Bernoulli’s principle. The low
pressure pulls up on the wing and pulls down on the air in accordance
with all of Newton’s laws.

See section 19.2 for additional discussion of how Newton’s
laws apply to the airplane and to the air.

The flow pattern created by a wing is the sum of the obstacle
effect (which is significant only very near the wing, and is the same
whether or not the wing is producing lift) plus the circulation effect
(which extends for huge distances above and below the wing, and is
proportional to the amount of lift, other things being equal).

A wing is very effective at changing the speed of the air.
The air above is speeded up relative to the corresponding air below.
Each air parcel gets a temporary change in speed and a permanent
offset in position.

Bernoulli’s principle asserts that a given parcel of air has
high velocity when it has low pressure, and vice versa. This is an
excellent approximation under a wide range of conditions. This can be
seen as a consequence of Newton’s laws.

Below-atmospheric pressure above the wing is
much more pronounced than above-atmospheric pressure below the
wing.

There is significant upwash ahead of the wing
and even more downwash behind the wing.

The front stagnation line is well below and behind
the leading edge.

The rear stagnation line is at or very near the
trailing edge. The Kutta condition says the air wants to flow
cleanly off the sharp trailing edge. This determines the amount
of circulation.

An airfoil does not have to be curved
on top and/or flat on the bottom in order to work. A rounded
leading edge is a good idea, but even a barn door will fly.

Air passing above and below the wing does not do so in
equal time. When lift is being produced, every air parcel passing
above the wing arrives substantially early (compared to
corresponding parcel below the wing) even though it has a longer path.

Most of the air above the wing arrives early in absolute terms
(compared to undisturbed air), but this is not important, and the
exceptions are doubly unimportant.

Lift is equal to circulation, times airspeed,
times density, times wingspan.

Well below the stalling angle of attack, the
coefficient of lift is proportional to the angle of attack; the
circulation is proportional to the coefficient of lift times the
airspeed.

Air is a fluid, not a bunch of bullets. The fluid has
pressure and velocity everywhere, not just where it meets the surface
of the wing.

There is downward momentum in any air column behind the wing.
There is zero momentum in any air column ahead of the wing, outboard
of the trailing vortices, or aft of the starting vortex.

Induced drag arises when you have low speed and/or short span,
because you are visiting a small amount of air and yanking it down
violently, producing strong wake vortices. In contrast there is very
little induced drag when you have high speed and/or long span,
because you are visiting a large amount of air, pulling it down
gently, producing weak wake vortices.

These simulations are based on a number of assumptions, including
that the viscosity is small (but not zero), the airspeed is small
compared to the speed of sound, the airflow is not significantly
turbulent, no fluid can flow through the surface of the wing, and the
points of interest are close to the wing and not too close to either
wingtip.

To be more precise: there is no wind in either of
the two dimensions that show up in figure 3.3.
There might be some flow in the third dimension (i.e. spanwise
along the stagnation line) but that isn’t relevant to the present
discussion.

This low pressure is associated
with fast-moving air in this region. You may be wondering why some of
this fast-moving air arrives at the trailing edge late. The answer is
that it spent a lot of time hanging around near the leading-edge
stagnation line, moving much slower than the ambient air. Then as it
passes the wing, it moves faster than ambient, but not faster enough
to make up for the lost time.

Of course, if there were no atmospheric pressure
below the wing, there would be no way to have reduced pressure above
the wing. Fundamentally, atmospheric pressure below the wing is
responsible for supporting the weight of the airplane. The point is
that pressure changes above the wing are more pronounced than the
pressure changes below the wing.

Actually, you
never get 100% of the circulation predicted by the Kutta condition,
especially for crummy airfoils like barn doors. For nice airfoils
with a rounded leading edge, you get something like 99% of the
Kutta circulation.

The second author’s name is properly spelled
Жуковский.
When Russian scientists write this name in English, they almost always
spell it Zhukovsky ... which is the spelling used in this book. Not
coincidentally, that conforms to standard transliteration rules and is
a reasonable guide to the pronunciation. Beware: you may encounter
the same name spelled other ways. In particular, “Joukowski” was
popular once upon a time, for no good reason.

This assumes the goal is to
produce wings, as opposed to (say) rudders. Also note that the
winglet solution may provide a practical advantage when taxiing and
parking. This is why Boeing put winglets (instead of additional span)
on the 747-400 — they wanted to be able to park in a standard slot
at the airport.

It’s only a
pseudo-tailwind, not a real tailwind, because wind is officially
supposed to be measured in the ambient air, someplace where the
air is not disturbed by the airplane — or by its mirror image.
Similarly airspeed is measured relative to the ambient air.